Problem: A small college has $800$ students, $10\%$ of which are left-handed. Suppose they take an SRS of $4$ students. Let $X=$ the number of left-handed students in the sample. What is the probability that exactly $2$ of the $4$ students are left-handed? You may round your answer to the nearest hundredth. $P(X=2)=$
Without a fancy calculator Having $2$ left-handed students in the sample of $4$ students means we need to have $2$ students that are left-handed and $2$ that aren't. We know $P({\text{left-handed}})={10\%}$ and $P({\text{not}})={90\%}$. We can assume independence since we are sampling less than $10\%$ of the population. So let's multiply probabilities to find the probability of getting $2$ left-handed students followed by $2$ students that aren't left-handed: $P({\text{LL}}{\text{NN}})=({0.10})^2({0.90})^2=0.0081$ This isn't our final answer, because there are other ways to get $2$ left-handed students in a sample of $4$ students (for example, NNLL). How many different ways are there? We can use the combination formula to find how many ways there are to get $2$ left-handed students in a sample of $4$ students: $\begin{aligned} _n\text{C}_k&=\dfrac{n!}{(n-k)!\cdot k!} \\\\ _4\text{C}_2&=\dfrac{4!}{(4-2)!\cdot2!} \\\\ &=\dfrac{4 \cdot 3 \cdot \cancel{2 \cdot 1}}{(2 \cdot 1) \cdot \cancel{2 \cdot 1}} \\\\ &=6 \end{aligned}$ There are $6$ ways to get $2$ left-handed students in a sample of $4$ students. Do they all have the same probability? Each of the $6$ ways has the same probability that we already found: $\begin{aligned} P({\text{LL}}{\text{NN}})&=({0.10})^2({0.90})^2=0.0081 \\\\ P({\text{L}}{\text{N}}{\text{L}}{\text{N}})&=({0.10})^2({0.90})^2=0.0081 \\\\ \vdots \\\\ P({\text{NN}}{\text{LL}})&=({0.10})^2({0.90})^2=0.0081 \end{aligned}$ So we can multiply this probability by $6$ since that is how many ways there are to get $2$ left-handed students in a sample of $4$ students: $\begin{aligned} P(X=2)&=6(0.10)^2(0.90)^2 \\\\ &=6(0.0081) \\\\ &=0.0486 \\\\ &\approx0.05 \end{aligned}$ Answer $P(X=2)=0.0486\approx0.05$